Let $T$ be the right shift operator, that is, $T(x_1,x_2,\ldots)=(0,x_1,x_2,\ldots)$, suppose $E$ is a compact operator from $\ell^{\infty}\rightarrow \ell^{\infty}$, can $T+E$ be an invertible operator from $\ell^{\infty}\rightarrow \ell^{\infty}$? It seems that $T+E$ can not be invertible, but I could not prove this.
We know that $T$ and $E$ are not surjective operators, I tried to show that $T+E$ is not surjective so that it can not be invertible but failed. Could some give any comments?
Best Answer
$T+E$ will be a Fredholm operator with index -1, so it will not be invertible.